Mr Brown and wife Sarah, whose daughter Jennifer died after she was born prematurely, have two sons, John and Fraser. Mr Brown said: "The day after my son was born I picked up the telephone and it was Nelson Mandela on the phone to congratulate me on the birth.
"We had both lost children and we had what was not a statesman to statesman conversation, but a father to father conversation.
"That was Mandela all the time - the personal care he took in his relationships, his great sense of humour, and his love of gossip and his love of talking about people, he was just a very complete and warm human being."
Mr Brown was premier in 2007 when Mandela and his wife Graca Machel attended the unveiling of a 9ft bronze statue of the ex-president opposite Parliament. Mandela was visibly moved as Mr Brown described the artwork as a "beacon of hope" for the oppressed around the world.
Mr Brown said last night: "What motivated him and drove him to risk his life for freedom was a burning passion that irrespective of colour, race and background, all people are created equal - and his list of historic achievements starts with a multi-racial South Africa.
"Every accolade in the world was awarded to him but the one he prized most was Children's Champion. As he said in his book, he had climbed one mountain, but there is another still to climb - dignity for every child. He was the greatest of Africans."